A103182 Sliding numbers: totals, including repetitions, of sums r + s, r >= s, such that 1/r + 1/s = (r + s)/10^k for some k >= 0.

2, 7, 11, 20, 25, 29, 52, 65, 70, 101, 110, 133, 200, 205, 205, 250, 254, 290, 425, 502, 520, 641, 650, 700, 785, 925, 1001, 1010, 1100, 1258, 1330, 2000, 2005, 2050, 2050, 2225, 2500, 2504, 2540, 2900, 3157, 3445, 4025, 4250, 5002, 5020

Offset

1,1

Comments

The matching values of k are given by A103183.

Note that 1/20 + 1/50 = 0.070 --> 70 is a "sliding number" though 0.070 is usually written as 0.07. So any "sliding number" G produces an infinite chain of others: G*10, G*100, G*1000, etc. The first number in such a chain is called "primitive". [Note the sequence is not defined by way of decimal digits. The power-of-10 relationship merely makes it practical to illustrate examples using decimal fractions. - Peter Munn, Nov 29 2022]

Another characterization of the sliding numbers is numbers of the form j + 10^k/j where j | 10^k (k >= 0). - David W. Wilson, Mar 09 2005

In the definition 1/r + 1/s is (r+s)/(r*s) and we can cancel out (r+s), so we are looking for all (r+s) with r and s positive integers, such that r*s is a (nonnegative integral) power of 10, sorted by (r+s). It follows, for example, that for a given k, the smallest (r+s) that is a solution is at least 2*sqrt(10^k) (equality when k is even, unless we also require r != s). - Christopher Landauer, Mar 10 2005

Equivalently, the terms are numbers with a partition into 2 integers whose product is a power of 10. - Peter Munn, Nov 28 2022

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Giovanni Resta, Sliding numbers

Example

1/4 + 1/25 = 0.29 --> 29 is a "sliding number"
1/8 + 1/125 = 0.133 --> 133 is a "sliding number"
1/2 + 1/5 = 0.7 --> 7 is a "sliding number"
From Hugo Pfoertner and Michael De Vlieger, Nov 28 2022: (Start)
Terms are repeated if they can be represented in different ways, for example:
205 = 5 + 200; 5 * 200 = 1000 (so 1/5 + 1/200 = 205/1000)
    = 80 + 125; 80 * 125 = 10000 (so 1/80 + 1/125 = 205/10000).
(End)

Mathematica

nn = 6; f[n_] := Transpose@ {Reverse@ #[[1 ;; Ceiling[Length[#]/2]]], #[[-Ceiling[Length[#]/2] ;; -1]]} &@ Divisors[10^n]; TakeWhile[Sort@ Flatten@ Table[Total /@ f[k], {k, 0, nn}], # < Total@ First@ f[nn + 1] &]] (* Michael De Vlieger, Oct 08 2022 *)

Crossrefs

Cf. A103183, A103184.

Sequence in context: A139603 A141183 A308724 * A358703 A160698 A294114

Adjacent sequences: A103179 A103180 A103181 * A103183 A103184 A103185

Keyword

nonn,base

Author

Eric Angelini, Mar 09 2005

Extensions

More terms from Kerry Mitchell, Mar 09 2005

Edited by Peter Munn, Nov 27 2022

Status

approved